Consider a feedforward network of single-server stations populated by multiple job types. Each job requires the completion of a number of tasks whose order of execution is determined by a set of deterministic precedence constraints. The precedence requirements allow some tasks to be done in parallel (in which case tasks would "fork") and require that others be processed sequentially (where tasks may "join"). Jobs of a given type share the same precedence constraints, interarrival time distributions and service time distributions, but these characteristics may vary across different job types. We show that the heavy traffic limit of certain processes associated with heterogeneous fork-join networks can be expressed as a semimartingale reflected Brownian motion with polyhedral state space. The polyhedral region typically has many more faces than its dimension, and the description of the state space becomes quite complicated in this setting. One can interpret the proliferation of additional faces in heterogeneous fork-join networks as (i) articulations of the fork and join constraints and (ii) consequences of the disordering effects that occur when jobs fork and join in their sojourns through the network.
"The Trouble with Diversity: Fork-Join Networks with Heterogeneous Customer Population." Ann. Appl. Probab. 4 (1) 1 - 25, February, 1994. https://doi.org/10.1214/aoap/1177005198